Known MEMS devices (Micro ElectroMechanical System devices) are integrated circuit structures wherein part of circuit structure has been undercut so that it is free to move relative to the underlying substrate. In addition the MEMS device contains electronic circuitry to generate electric fields to exert forces on the free part of the structure and/or measure electric parameters relating to the free part, such as current flowing to it, accumulated charge etc. In a typical MEMS device the free part is made of crystalline silicon, but other materials may be used, dependent on the technology used to manufacture the integrated circuit.
Devices with micromechanical structures are typically manufactured by processing a silicon wafer, using techniques known from integrated circuit processing. In order to create a free part in a micro mechanical structure a two-dimensional shape of a mechanical bar or block on the wafer may be defined by lithography, and undercut etching may be used to detach part of the bar or block from the underlying material. Thus it is possible to create a bar or block of crystalline silicon material that is free to vibrate mechanically over at least part of its length. Moreover, during manufacture of such a device electrodes may be defined that can be used to apply electrical fields in order to excite the vibrations.
PCT patent application WO2006/067757 describes a charge biased micro-mechanical resonator, containing a micromechanical structure. This document shows a clamped bar structure, which comprises an elongated strip of material, with end portions attached to the underlying substrate and an unattached mid portion that is free to move subject to the laws of elasticity that govern deformation of the bar. An electrode next to this mid portion can be used to excite vibrations.
Non-linearity is a known problem of this type of device. For small deformation the behavior of the structure is linear, i.e. the amount of deformation is proportional to force applied to the structure. For larger deformations non-linearity gives rise to deviations from the linear behavior. Non-linearity may have detrimental effects in terms of decreased power efficiency, lower resonance quality etc. Non-linearity arises, among others, due to non-linear elastic properties of the material of the structure. WO2006/067757 describes that the use of stored charge can reduce non-linearity.
Another solution to the problem of non-linearity is described in an article by Manu Agarwal et al. in Applied Physics Letters 89, 214105 2006, pages 214105-1 to 214105-3 and titled “Optimal drive condition for nonlinearity reduction in electrostatic microresonators”. Agarwal discloses that in a clamped bar the non linearity due to the mechanical properties of the bar can be compensated at certain excitation amplitudes by opposite non-linear behavior of the electrical field strength of the field from the electrode. With a fixed voltage difference between the electrode and the bar, the force due to this field is inversely proportional to the square of the distance between the bar and the electrode, which means that the force depends non-linearly on the displacement of the bar. Agarwal et al note that this non-linearity can compensate the elastic non-linearity. However, this solution only works over a narrow range of excitation amplitudes, as the non-linearity of the electric field and the non-linearity due to the material properties and geometry depend differently on amplitude. Moreover, the double clamping of the bar leads to energy losses which reduce the quality factor of the resonance.